A committee of 3 women and 4 men is to be chosen randomly from a group of 20 wom

Question

A committee of 3 women and 4 men is to be chosen randomly from a group of 20 women and 18 men.

(a) (5 points) What is the probability that “Ann”(female) and her friend “Bob”(male) are on the committee together?

(b) (5 points) What is the probability that “Ann”(female) and her friend “Bob”(male) are on the committee and their friends “Carla”(female) and “Dan”(male) are also on the committee?

(c) (5 points) Given that “Carla” and “Dan” have been chosen to be on the committee, what is the probability that “Ann” and her friend “Bob” are chosen to be on committee?

(d) (5 points) What is the probability that none of “Ann”,“Bob”,“Carla”, or “Dan” are chosen to be on the committee?

Explanation / Answer

Total number of people: 20+18 = 38

Total number of ways of choosing 3 women and 4 men is

C(20,3) * C(18,4) = 3488400

(a)

After choosing Ann”(female) and her friend “Bob”(male) on the committee together, number of ways of choosing remaining members is

C(19,2)*C(17,3) = 116280

So, the probability that “Ann”(female) and her friend “Bob”(male) are on the committee together is

116280 / 3488400 = 0.0333

(b)

After choosing Ann”(female) and her friend “Bob”(male) and their friends “Carla”(female) and “Dan”(male) on the committee together, number of ways of choosing remaining members is

C(18,1)*C(16,2) = 2160

So, the probability that “Ann”(female) and her friend “Bob”(male) are on the committee and their friends “Carla”(female) and “Dan”(male) are also on the committee is

2160 / 3488400 = 0.00062

(c)

The probability that “Carla” and “Dan” have been chosen to be on the committee is

116280 / 3488400 = 0.0333

So the probability that “Ann” and her friend “Bob” are chosen to be on committee given that “Carla” and “Dan” have been chosen to be on the committee is

P( “Ann” and her friend “Bob” | "Carla” and “Dan”) = P( “Ann” and her friend “Bob” and "Carla” and “Dan”) / P("Carla” and “Dan”) = (2160 / 3488400 ) / (116280 / 3488400) = 0.0186

(d)

Number of ways of choosing 3 women and 4 men from remaining 18 women and 16 men is

C(18,3) * C(16,4) = 8160

So, the probability that none of “Ann”,“Bob”,“Carla”, or “Dan” are chosen to be on the committee is

8160 / 3488400 = 0.0023

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